Question: The Jacksons and the Simpsons were competing in the final leg of the Amazing Race, which was $240$ kilometers long. In their race to the finish, the Jacksons immediately took off traveling at an average speed of $v$ kilometers per hour. The Simpsons' start was delayed by an hour. When they eventually took off, they traveled at an average speed that was $40$ kilometers per hour faster than the Jacksons' speed. Sadly for them, that didn't help, and the Jacksons won. Write an inequality in terms of $v$ that models the situation.
Solution: The strategy In the final leg of the Amazing Race, the team that arrives first wins the race. Since the Jacksons win the race, it takes them less time to complete the race than the Simpsons. If we let $J$ be the time it takes for the Jacksons to complete the race and $S$ be the time it takes for the Simpsons to complete the race, we have that $J<S$. Now, let's express $J$ and $S$ in terms of $v$. Expressing the time it takes for the Jacksons to complete the race We know that $\text{distance}=\text{speed}\cdot \text{time}$ and so $\text{time}=\dfrac{\text{distance}}{\text{speed}}$. Since the Jacksons start immediately and travel $240$ kilometers at an average speed of $v$ kilometers per hour, it took them $\dfrac{240}{v}$ hours to complete the race. Expressing the time it takes for the Simpsons to complete the race We know that the Simpsons' average speed was $40$ kilometers per hour faster than the Jacksons' average speed, or $v+40$ kilometers per hour. Since the Simpsons traveled $240$ kilometers at an average speed of $v+40$ kilometers per hour, it took them $\dfrac{240}{v+40}$ hours to complete the race. However, they were delayed by $1$ hour, so their total time was $\dfrac{240}{v+40}+1$ hours. Putting things together We found that $J=\dfrac{240}{v}$ and $S=\dfrac{240}{v+40}+1$. Since $J<S$, we can substitute and find an inequality in terms of $v$ that models the situation. The answer is: $ \dfrac{240}{v}<\dfrac{240}{v+40}+1$